Problem: Given $ \overrightarrow{PQ}\perp\overrightarrow{PS}$, $ m \angle QPR = 3x + 19$, and $ m \angle RPS = 8x + 60$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Answer: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since we are given that $\overrightarrow{PQ}\perp\overrightarrow{PS}$ , we know ${m\angle QPS = 90}$ Substitute in the expressions that were given for each measure: $ {3x + 19} + {8x + 60} = {90}$ Combine like terms: $ 11x + 79 = 90$ Subtract $79$ from both sides: $ 11x = 11$ Divide both sides by $11$ to find $x$ $ x = 1$ Substitute $1$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 8({1}) + 60$ Simplify: $ {m\angle RPS = 8 + 60}$ So ${m\angle RPS = 68}$.